The Lagrangian fibration \(\pi \,:\,\mathbb {C}^2\rightarrow \mathbb {C}\)  is by the level sets \(p^2-u(q)=h_2\) . Singular fibers correspond to the stationary values of the Hamiltonian, i.e. to the equilibria \((p,q)=(0, q_0)\)  with \(u^{\prime }(q_0)=0\) .
Then we can find that for a number of \(l \ge 1\)  generic couplings \(g_i\) , the number of stationary values of \(h_2\)  is \(l+2\) , in agreement with the Seiberg-Witten geometry [1]}. Indeed, the function \(u^{\prime }(q)= \sum _{i=0}^{3} g_i^2 \wp ^{\prime }(q+\omega _i)\)  is odd elliptic of order \(3l\) , therefore it has \(3l\)  zeros; the genericity assumption ensures that the multiplicity of each zero is always one. Then \(4-l\)  zeros are given by the half-periods, for which the values of \(h_2\)  are distinct. Furthermore, the other \(4l-4\)  zeros come in pairs \((q, -q)\)  so give the same stationary value of \(h_2\) . Thus, the number of singular fibers (or stationary values of \(h_2\) ) is \((4-l) + (4l-4)/2 = l+2 \) , as claimed.
